#283660

order	512
nilpotency class	3
isoclinism	isoclinic class of #283643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25616	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,9,7,4,5,14)(3,10,12,8,13,16,15,11)(17,18,20,23)(19,25,27,22)(21,24,28,31)(26,32,29,30);(1,3,5,15,7,13,6,12)(2,8,14,10,4,11,9,16)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(1,2)(3,11)(4,7)(5,9)(6,14)(8,13)(10,15)(12,16)(17,19)(18,22)(20,27)(21,29)(23,25)(24,32)(26,28)(30,31);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,15,16,12)(19,27)(22,25)(26,29)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,d^c=di,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 190856);